3,503 research outputs found
A Metric for genus-zero surfaces
We present a new method to compare the shapes of genus-zero surfaces. We
introduce a measure of mutual stretching, the symmetric distortion energy, and
establish the existence of a conformal diffeomorphism between any two
genus-zero surfaces that minimizes this energy. We then prove that the energies
of the minimizing diffeomorphisms give a metric on the space of genus-zero
Riemannian surfaces. This metric and the corresponding optimal diffeomorphisms
are shown to have properties that are highly desirable for applications.Comment: 33 pages, 8 figure
Quasiconvex Programming
We define quasiconvex programming, a form of generalized linear programming
in which one seeks the point minimizing the pointwise maximum of a collection
of quasiconvex functions. We survey algorithms for solving quasiconvex programs
either numerically or via generalizations of the dual simplex method from
linear programming, and describe varied applications of this geometric
optimization technique in meshing, scientific computation, information
visualization, automated algorithm analysis, and robust statistics.Comment: 33 pages, 14 figure
Grid generation for the solution of partial differential equations
A general survey of grid generators is presented with a concern for understanding why grids are necessary, how they are applied, and how they are generated. After an examination of the need for meshes, the overall applications setting is established with a categorization of the various connectivity patterns. This is split between structured grids and unstructured meshes. Altogether, the categorization establishes the foundation upon which grid generation techniques are developed. The two primary categories are algebraic techniques and partial differential equation techniques. These are each split into basic parts, and accordingly are individually examined in some detail. In the process, the interrelations between the various parts are accented. From the established background in the primary techniques, consideration is shifted to the topic of interactive grid generation and then to adaptive meshes. The setting for adaptivity is established with a suitable means to monitor severe solution behavior. Adaptive grids are considered first and are followed by adaptive triangular meshes. Then the consideration shifts to the temporal coupling between grid generators and PDE-solvers. To conclude, a reflection upon the discussion, herein, is given
Proceedings of the Fifth Workshop on Information Theoretic Methods in Science and Engineering
These are the online proceedings of the Fifth Workshop on Information Theoretic Methods in Science and Engineering (WITMSE), which was held in the Trippenhuis, Amsterdam, in August 2012
Conformal Wasserstein distances: comparing surfaces in polynomial time
We present a constructive approach to surface comparison realizable by a
polynomial-time algorithm. We determine the "similarity" of two given surfaces
by solving a mass-transportation problem between their conformal densities.
This mass transportation problem differs from the standard case in that we
require the solution to be invariant under global M\"{o}bius transformations.
We present in detail the case where the surfaces to compare are disk-like; we
also sketch how the approach can be generalized to other types of surfaces.Comment: 23 pages, 3 figure
New algorithmic developments in maximum consensus robust fitting
In many computer vision applications, the task of robustly estimating the set of parameters of
a geometric model is a fundamental problem. Despite the longstanding research efforts on robust
model fitting, there remains significant scope for investigation. For a large number of geometric
estimation tasks in computer vision, maximum consensus is the most popular robust fitting
criterion. This thesis makes several contributions in the algorithms for consensus maximization.
Randomized hypothesize-and-verify algorithms are arguably the most widely used class of
techniques for robust estimation thanks to their simplicity. Though efficient, these randomized
heuristic methods do not guarantee finding good maximum consensus estimates. To improve the
randomize algorithms, guided sampling approaches have been developed. These methods take
advantage of additional domain information, such as descriptor matching scores, to guide the
sampling process. Subsets of the data that are more likely to result in good estimates are prioritized
for consideration. However, these guided sampling approaches are ineffective when good
domain information is not available. This thesis tackles this shortcoming by proposing a new
guided sampling algorithm, which is based on the class of LP-type problems and Monte Carlo
Tree Search (MCTS). The proposed algorithm relies on a fundamental geometric arrangement
of the data to guide the sampling process. Specifically, we take advantage of the underlying tree
structure of the maximum consensus problem and apply MCTS to efficiently search the tree.
Empirical results show that the new guided sampling strategy outperforms traditional randomized
methods.
Consensus maximization also plays a key role in robust point set registration. A special case
is the registration of deformable shapes. If the surfaces have the same intrinsic shapes, their
deformations can be described accurately by a conformal model. The uniformization theorem
allows the shapes to be conformally mapped onto a canonical domain, wherein the shapes can be
aligned using a M¨obius transformation. The problem of correspondence-free M¨obius alignment
of two sets of noisy and partially overlapping point sets can be tackled as a maximum consensus
problem. Solving for the M¨obius transformation can be approached by randomized voting-type
methods which offers no guarantee of optimality. Local methods such as Iterative Closest Point
can be applied, but with the assumption that a good initialization is given or these techniques
may converge to a bad local minima. When a globally optimal solution is required, the literature
has so far considered only brute-force search. This thesis contributes a new branch-and-bound
algorithm that solves for the globally optimal M¨obius transformation much more efficiently.
So far, the consensus maximization problems are approached mainly by randomized algorithms,
which are efficient but offer no analytical convergence guarantee. On the other hand,
there exist exact algorithms that can solve the problem up to global optimality. The global methods,
however, are intractable in general due to the NP-hardness of the consensus maximization. To fill the gap between the two extremes, this thesis contributes two novel deterministic algorithms
to approximately optimize the maximum consensus criterion. The first method is based
on non-smooth penalization supported by a Frank-Wolfe-style optimization scheme, and another
algorithm is based on Alternating Direction Method of Multipliers (ADMM). Both of the
proposed methods are capable of handling the non-linear geometric residuals commonly used in
computer vision. As will be demonstrated, our proposed methods consistently outperform other
heuristics and approximate methods.Thesis (Ph.D.) (Research by Publication) -- University of Adelaide, School of Computer Science, 201
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